A211226 - OEIS

%I #9 Apr 11 2012 12:57:16

%S 1,1,1,1,1,1,1,1,1,1,1,2,2,2,1,1,1,2,2,1,1,1,3,3,6,3,3,1,1,1,3,3,3,3,

%T 1,1,1,4,4,12,6,12,4,4,1,1,1,4,4,6,6,4,4,1,1,1,5,5,20,10,30,10,20,5,5,

%U 1,1,1,5,5,10,10,10,10,5,5,1,1,1,6,6,30,15

%N Triangular array: T(n,k) = f(n)/(f(k)*f(n-k)), where f(n) = (floor(n/2))!.

%H Peter Bala, <a href="/A211226/a211226.txt">Notes on A211226</a>

%F T(n,k) := f(n)/(f(k)*f(n-k)), where f(n) := (floor(n/2))!.

%F T(2*n+1,2*k) = T(2*n+1,2*k+1) = T(2*n,2*k) = binomial(n,k);

%F T(2*n,2*k+1) = n*binomial(n-1,k).

%F Recurrence equations:

%F T(2*n,2*k) = T(2*n-1,2*k) + T(2*n-1,2*k-1);

%F T(2*n,2*k+1) = T(2*n-1,2*k+1) + (n-1)*T(2*n-1,2*k);

%F T(2*n+1,2*k) = T(2*n,2*k); T(2*n+1,2*k+1) = T(2*n,2*k).

%F The Star of David property holds:

%F T(n,k+1)*T(n+1,k)*T(n+2,k+2) = T(n,k)*T(n+2,k+1)*T(n+1,k+2).

%F O.g.f.: (1 + t*(1+x) - t^2*(1-x+x^2) - t^3*(1+x+x^2+x^3))/(1 - t^2*(1+x^2))^2 = sum {n>=0} R(n,x)*t^n = 1 + (1+x)*t + (1+x+x^2)*t^2 + (1+x+x^2+x^3)*t^3 + ....

%F E.g.f.: cosh(t*sqrt(1+x^2)) + (1+x+x*t/2)/sqrt(1+x^2)*sinh(t*sqrt(1+x^2)) = sum {n>=0} R(n,x)*t^n/n! = 1 + (1+x)*t + (1+x+x^2)*t^2/2! + (1+x+x^2+x^3)*t^3/3! + ....

%F Row generating polynomials: R(2*n+1,x) = (1+x)*(1+x^2)^n; R(2*n,x) = (1+n*x+x^2)*(1+x^2)^(n-1).

%F Row sums: A211227. Shallow diagonal sums A211228. Central terms T(2*n,n) equal A056040(n).

%F The inverse array A211229 involves the derangement numbers A000166. The squared array is A211230.

%e Triangle begins

%e .n\k.|....0....1....2....3....4....5....6....7....8....9...10...11

%e = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = =

%e ..0..|....1

%e ..1..|....1....1

%e ..2..|....1....1....1

%e ..3..|....1....1....1....1

%e ..4..|....1....2....2....2....1

%e ..5..|....1....1....2....2....1....1

%e ..6..|....1....3....3....6....3....3....1

%e ..7..|....1....1....3....3....3....3....1....1

%e ..8..|....1....4....4...12....6...12....4....4....1

%e ..9..|....1....1....4....4....6....6....4....4....1....1

%e .10..|....1....5....5...20...10...30...10...20....5....5....1

%e .11..|....1....1....5....5...10...10...10...10....5....5....1....1

%e ...

%Y Cf. A007318, A056040, A211227 (row sums), A211228 (shallow diagonal sums), A211229 (inverse), A211230 (array squared).

%K nonn,easy,tabl

%O 0,12

%A _Peter Bala_, Apr 05 2012